Training Manual on Transport and Fluids *OHN#.EU 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/109 Training Manual on Transport and Fluids Training Manual on Transport and Fluids John C. Neu Graduate Studies in Mathematics Volume 109 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 35–XX, 44–XX, 76–XX. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-109 Library of Congress Cataloging-in-Publication Data Neu, John C., 1952– Training manual on transport and fluids / John C. Neu. p. cm. — (Graduate studies in mathematics ; v. 109) ISBN 978-0-8218-4083-2 (alk. paper) 1. Transport theory—Mathematical models. 2. Fluid mechanics—Mathematical models. I. Title. QC718.5.T7N48 2998 530.138—dc22 2009028296 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface xi Part 1. Transport processes: the basic prototypes Chapter 1. Convection 3 Material volumes and transport theorem 3 Problem 1.1: Convection along a one-dimensional line 5 Problem 1.2: Jacobian of flow map 6 Problem 1.3: Transport theorem by change of variable 8 Problem 1.4: Time derivative of integral over a changing region 9 The convection PDE 10 Problem 1.5: Density carried by incompressible flow 10 Problem 1.6: Transformation of velocities as a flow map 11 Convective flux and boundary conditions 12 Elastic rebounds from a moving wall 13 Problem 1.7: Adiabatic expansion of particles in a box I 15 Problem 1.8: Adiabatic expansion of particles in a box II 15 Problem 1.9: Adiabatic expansion of particles in a box III 16 Problem 1.10: Adiabatic expansion of particles in a box IV 17 Convective derivative 18 Convected scalars 19 Problem 1.11: A convected scalar 19 Gradient of a convected scalar 20 Problem 1.12: Flows that preserve gradients 22 Problem 1.13: Vector analog of a convected scalar 24 Problem 1.14: Vector analogs of a convected density 25 Geometric postscript 29 v vi Contents Guide to bibliography 30 Chapter 2. Diffusion 31 Brownian motion 31 The diffusion PDE 33 Problem 2.1: Lattice random walk model of diffusion I 35 Problem 2.2: Lattice random walk model of diffusion II 36 Problem 2.3: Lattice random walk model of diffusion III 37 Problem 2.4: Lattice random walk model of diffusion IV 38 Problem 2.5: Lattice random walk model of diffusion V 38 Diffusion in Rn 39 Fick’s law 39 Problem 2.6: Lyapunov functional for a diffusion process 40 Problem 2.7: The Maxwell flux 41 Problem 2.8: Steady state in two dimensions? 42 Melting interfaces 42 Problem 2.9: Nonlinear free boundary problem I 45 Problem 2.10: Nonlinear free boundary problem II 46 Problem 2.11: Nonlinear free boundary problem III 47 Problem 2.12: Nonlinear free boundary problem IV 48 The diffusion kernel 49 Scale invariance of the diffusion kernel 50 Problem 2.13: R2 diffusion kernel from symmetry and independence 53 Problem 2.14: Similarity solution for melting interfaces 54 Problem 2.15: Source of Brownian particles in two dimensions 56 Problem 2.16: Homogeneous harmonic function in a wedge 58 Guide to bibliography 59 Chapter 3. Local conservation laws 61 Definition 61 Convection-diffusion 61 Problem 3.1: A “dissolving” interface 62 Smoluchowski boundary value problem 65 Problem 3.2: Time-independent equilibria 67 Problem 3.3: Entropy functional 68 Problem 3.4: Free energy decreases 70 Problem 3.5: Equilibrium or nonequilibrium in Coulomb potential 70 Locality 72 Sources 75 The steady diffusion flame 76 Problem 3.6: Absorbing boundary condition for diffusion 80 Problem 3.7: An “evaporation” interface driven by chemical reaction 82 Contents vii Guide to bibliography 84 Part 2. Superposition Chapter 4. Superposition of point source solutions 87 One space dimension 87 Problem 4.1: Soviet rod factory 89 Three space dimensions 91 Problem 4.2: Superposition solution by Green’s identity 93 Problem 4.3: The smoke plume 94 Problem 4.4: Far downstream limit 96 Sources in spacetime 98 Problem 4.5: Moving source 101 The hottest time of the day 102 Problem 4.6: “Diffusive” damping 105 Problem 4.7: Work against “diffusive friction” 107 Problem 4.8: Oscillatory point source solution in R3 and descent to R2 109 Problem 4.9: Direct analysis of the R2 integral representation 111 Guide to bibliography 113 Chapter 5. δ-functions 115 δ-functions as “shorthand” 116 Problem 5.1: Correctly representing a point load 118 Problem 5.2: Correlation function of colored noise 120 Problem 5.3: A primer on polarized media 125 Problem 5.4: Exit times for Brownian motion 128 δ-sequences 130 Problem 5.5: Green’s identity and inhomogeneous diffusion PDEs 132 “Oscillatory” δ-sequences 134 Problem 5.6: Cancellation by oscillation 136 Problem 5.7: An oscillatory δ-sequence 138 Guide to bibliography 140 Part 3. Scaling-based reductions in basic fluid mechanics Chapter 6. Ideal fluid mechanics 143 Ideal fluid 143 Problem 6.1: Tornado season 147 Problem 6.2: Ideal gas equation of state 148 Problem 6.3: Bernoulli’s theorem and flying over Dagoba 150 Problem 6.4: Circulation and vorticity 152 Problem 6.5: Heating due to shock wave 153 viii Contents Stable rest state of ideal fluid 157 Finite propagation speed and wavefronts 158 Problem 6.6: Sonic boom 160 Incompressible fluid 165 Problem 6.7: Two-dimensional vortex dynamics 166 Problem 6.8: The “ground effect” 169 Problem 6.9: Elliptical vortices 172 The incompressible limit of compressible flow 176 Problem 6.10: Buoyancy force 178 Problem 6.11: Internal waves 181 Guide to bibliography 184 Chapter 7. Free surface waves 187 Basic equations 187 Linearized waves 189 Plane waves and the dispersion relation 190 Problem 7.1: Internal waves, again 193 Shallow water equations 197 Problem 7.2: Variable-depth ocean and “slow currents” 200 Problem 7.3: What bottom topography does to vortices 202 Problem 7.4: A cautionary tale about point vortices 204 Guide to bibliography 207 Chapter 8. Solution of the shallow water equations 209 Simple waves 209 Problem 8.1: Simple waves in ideal fluid 212 Problem 8.2: Longitudinal elastic waves 215 Characteristics and Riemann invariants 219 Problem 8.3: Characteristics and Riemann invariants for elastic waves 222 Problem 8.4: Beginning of a tsunami 224 Dam break problem 227 Problem 8.5: Wave generator 228 Bores 231 Problem 8.6: “Potential energy” in shallow water theory 234 Problem 8.7: Elastic shocks 235 Characteristics “eaten” by a bore 238 The afterlife of a simple wave 239 Propagation of a weak bore 240 Problem 8.8: Area rule for Burger’s shocks 242 Problem 8.9: Traveling waves in variable-depth ocean 246 Problem 8.10: Tsunami reveals its true self 251 Guide to bibliography 255 Contents ix Bibliography 257 Index 259 Preface This text presents models of transport in continuous media and a corre- sponding body of mathematical techniques. Within this text, I have em- bedded a subtext of problems. Topics and problems are listed together in the table of contents. Each problem is followed by a detailed solution em- phasizing process and craftsmanship. These problems and solutions express the practice of applied mathematics as the examination and re-examination of essential ideas in many interrelated examples. Since the science that falls under the headings “transport” or “fluids” is so broad, this introductory text for a one-term advanced undergraduate or beginning graduate course must select a highly specific path. The main requirement is that topics and exercises be logically interconnected and form a self-contained whole. Briefly, the physical topics are: convection and diffusion as the sim- plest models of transport, local conservation laws with sources as a general “frame” of continuum mechanics, ideal fluid as the simplest example of an actual physical medium with mass, momentum and energy transport, and finally, free surface waves and shallow water theory. The idea behind this lineup is the progression from purely geometric and kinematic to genuinely physical. The mathematical prerequisites for engaging the practice of this text are: fluency in advanced calculus and vector analysis, and acquaintance with PDEs from an introductory undergraduate course. The mathematical skills developed in this text have two tracks: First, classical constructions of solutions to linear PDEs and related tools, such as the Dirac δ-function, are presented with a relentless sense of connection to xi xii Preface the geometric-physical situations they articulate. Second, and more essen- tial, is the emphasis on dimensional analysis and scaling.